Let $\mathrm{A} = \left[\begin{array}{ccc}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{h}& \mathrm{b}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right] \mathrm{and} \mathrm{B} = \left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right] \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{AB} = ?$

• $\left[\begin{array}{c}\mathrm{ax} + \mathrm{hy} + \mathrm{gz}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$

• a + b

• $\left[\begin{array}{c}\mathrm{ax} + \mathrm{hy} + \mathrm{gz}\\ \mathrm{hx} + \mathrm{by} + \mathrm{fz}\\ \mathrm{gx} + \mathrm{fy} +\hspace{0.17em}\mathrm{cz}\end{array}\right]$

• $\left[\begin{array}{ccc}\mathrm{ax} + \mathrm{hy} +\mathrm{gz}& \mathrm{hz} + \mathrm{by} + \mathrm{fz}& \mathrm{gx} + \mathrm{fy} + \mathrm{cz}\end{array}\right]$

For how many values of k, is the matrix $\left[\begin{array}{ccc}0& \mathrm{k}& 4\\ - \mathrm{k}& 0& - 5\\ - \mathrm{k}& \mathrm{k}& - 1\end{array}\right]$

singular ?

• only one

• Only two

• Only three

• Infinite

43.

If matrix $\left[\begin{array}{cc}1 - \mathrm{i}& \mathrm{i}\\ - \mathrm{i}& 1 - \mathrm{i}\end{array}\right]$ where $\mathrm{i} =\sqrt{ - 1}$, then which one of the following is correct ?

• A is hermitian

• A is skew-hermitian

• ${\left(\overline{\mathrm{A}}\right)}^{\mathrm{T}} + \mathrm{A}$ is hermitian

• ${\left(\overline{\mathrm{A}}\right)}^{\mathrm{T}} +\hspace{0.17em}\mathrm{A}$is skew-hermitian

If A is a matrix of order 3 × 5 and B is a matrix of order 5 × 3, then the order of AB and BA will respectively be

• 3 × 3 and 3 × 3

• 3 × 5 and 5 × 3

• 3 × 3 and 5 × 5

• 5 × 3 and 3 × 5

Under which condition, are the points (a, b), (c, d) and (a – c, b - d) collinear ?

• ab = cd

• ac = bd

• ad = bc

• abc = d

Consider the following in respect of a non-singular matrix of order

1) A (adj A) = (adj A) A

2) |adj A| = |A|

Which of the above statements is/are correct ?

• 1 only

• 2 only

• Both 1 and 2

• Neither 1 nor 2